> > > If R is a finite commutative ring without multiplicative identity> > > and if every element is a zero divisor, then does there exist> > > a nonzero element which annihilates all elements of the ring?> > Ask-an-Algebraist> > No - the trivial ring. > > Incorrect. The trivial ring *does* have a multiplicative identity. I'll let you figure out what it is and why it *does* satisfy the condition that 1x=x=x1 for all x in the ring.> In fact, I'll give you three guesses.> The first two don't count, though.> Your trivial ring isn't as trivial as my trivial ring because your trivial ring is fancied up with a multiplicative identity.

> > So add the premise that R has a nonzero element.> Or, perhaps, not.

Definitely so for OP asked about rings without multiplicative identities.